Sanderson M. Smith

DECCO (CALIFORNIA LOTTERY GAME... GOOD PROBABILITY MATERIAL)

I must complain the cards are ill-shuffled 'til I have a good hand.
-Swift, Thoughts on Various Subjects

The California Lottery game of DECCO is fairly simple. You pay \$1 to play. You simply choose one card from each suit in a standard deck of 52 cards. For instance, you might choose the ace of hearts, the ten of clubs, the jack of diamonds, and the three of spades. The State of California then randomly generates four cards, one from each suit.

If you match all four cards, you win \$500.
If you match
three of the four cards, you win \$50.
If you match
two of the four cards, you win \$5.
If you match
one of the four cards, you get a replay ticket. (This is not equivalent to getting your dollar back.)
If you
don't match any of the four cards, you lose your dollar.

Some DECCO-related calculations:

There are 134 = 28,561 possible DECCO hands.

The number of ways you can match...

4 cards is (4C4)(1C1)4 = 1

3 cards is (4C3)(1C1)3(12C1) = 48

2 cards is (4C2)(1C1)2(12C1)2 = 864

1 card is (4C1)(1C1)(12C1)3 = 6,912

0 cards is (4C0)(12C1)4 = 20,736

If x is not zero, the algebraic identity x = 1/(1/x) is frequently useful in expressing probabilities. For instance, if the probability of an event is 0.000085 (or 0.0085%), it would generally be considered to be an event not likely to happen. However, one might like to know "how unlikely?" OK, we have 0.000085 = 1/(1/0.000085) = 1/11765. In other words, one would expect the event to happen once in every 11,765 trials. This algebraic identity is used in the following DECCO-related table.

 EVENT # ways event can happen probability Expected State Gain in 28,561 games Expected State Loss in 28,561 games Match 4 1 1/28561 \$1 \$5,000 Match 3 48 48/28561 = 1/595 \$48 \$2,400 Match 2 864 864/28561 = 1/33 \$864 \$4,320 Match 1 6,912 6912/28561 = 1/4.13 \$6,912 \$3,742 Match 0 20,736 20736/28561 = 1/1.38 \$20,736 \$0 TOTALS 28,561 100% \$28,561 \$15,462

Summary (from table): In 28,561 games...

California expects to take in \$28,561

California expects to pay out \$15,462

California expected gain in 28,561 games \$13,099

California expected gain per gain \$0.4586

In other words, for every \$1 spent to play DECCO, California expects to gain about 46 cents.

Now, the expected state loss for the "match 1 situation" probably requires some explanation. When you match 1 card, you get a free replay ticket. Some folks think this is equivalent to getting your dollar back, but this is not true. In one sense, you never get your dollar back. You must always end up in one of the other matching categories. You might eventually lose the dollar, or you might win money as a result of a replay, but you do not literally get your dollar back. You will eventually end up in one of the other four categories. Referencing the table, the ratio of occurrence for the other categories is

(Match 4):(Match 3):(Match 2):(Match 0) = 1:48:864:20,736.

Now, 1+48+864+20,736 = 21,649.

Hence, if you initially match 1 card, the probability that you will end matching

4 cards is 1/21649 = 0.00% (rounded to two decimals)

3 cards is 48/21649 = 0.22%

2 cards is 864/21649 = 3.99%

0 cards is 20736/21694 = 95.78%

So, in 28,561 games, the expected State loss for the "match 1 card" event is

6912[\$5000/21649 + \$50(48)/21649 + \$5(864)/21649 + \$0(20736/21694] = \$3,742

the figure that appears in the table.

To repeat, the statistical conclusion is that the State of California got about 46 cents out of every dollar that was spent on DECCO. So, when you spend \$1 to play DECCO, you are basically contributing 46 cents to the State of California. We are told that 34% of the money made from lottery games benefits California education, so an "investment" of \$1 in Decco results in a donation of about 16 cents for educational purposes in the State.